Exponents and logarithms: exam review Exam review: second paper One question at a time

Use the calculator where it helps. [7]

1. (a) Solve \(10^x = 3600\). Give x to 3 significant figures. [2]

   • M1 for x = log10 3600
   • A1 3.56

2. (b) Solve \(\log_{10}(4x + 1) = 2\). Give the exact value of x. [2]

   • M1 for 4x + 1 = 10^2 = 100
   • A1 x = \frac{99}{4} (24.8)

3. (c) Solve \(2 + 4\ln x = 11\). Give the exact value of x, and x to 3 significant figures. [3]

   • M1 for isolating the log: ln x = \frac{9}{4}
   • M1 for x = e^(\frac{9}{4})
   • A1 9.49

Do not use a calculator for this question. [5]

1. (a) Write \(\dfrac{(2x^5)^3}{12x^{-4}}\) in the form \(ax^p\). Write down the values of a and p. [3]

   • M1 for (2x^5)^3 = 8x^15
   • M1 for dividing by x^(−4), so the indices are added: 15 + 4
   • A1 a = 2/3, p = 19

2. (b) Find the exact value of \(\left(\dfrac{2}{5}\right)^{-3}\). [2]

   • M1 for turning the fraction over: (5/2)^3
   • A1 125/8

Do not use a calculator for this question. [7]

1. (a) Find the exact value of \(64^{-\frac{2}{3}}\). [2]

   • M1 for 64^(1/3) = 4
   • A1 1/16

2. (b) Given that \(y=\dfrac{3}{x^3\sqrt{x}}\), write \(y^2\) in the form \(ax^k\). Write down the values of a and k. [3]

   • M1 for writing √x as x^(1/2), so y = 3x^(−3.5)
   • M1 for squaring, multiplying the index by 2
   • A1 a = 9, k = −7

3. (c) Write \(\dfrac{x^3 + 4\sqrt{x}}{x}\) in the form \(ax^p + bx^q\). [2]

   • M1 for dividing each term by x
   • A1 x^2 + 4x^(−1/2)

Solve each equation. Do not use a calculator. [5]

1. (a) Solve \(x^{\frac{3}{2}} = 125\). [2]

   • M1 for raising both sides to the power 2/3: x = 125^(2/3)
   • A1 x = 25

2. (b) Solve \(\left(\sqrt{3}\right)^x = 27^{x-2}\). Give the exact value of x. [3]

   • M1 for writing both sides as powers of 3: 3^(x/2) = 3^(3(x − 2))
   • M1 for equating the indices: x/2 = 3x − 6
   • A1 x = 12/5

Give exact answers. Do not use a calculator. [5]

1. (a) Let \(p=\log_{10}a\) and \(q=\log_{10}b\). Write \(\log_{10}\left(\dfrac{1000\sqrt{a}}{b^{2}}\right)\) in terms of p and q. [3]

   • M1 for the product or quotient law: log of a product is a sum, log of a quotient is a difference
   • M1 for the power law, including the root as a power 1/2
   • A1 3 + \frac{1}{2}p − 2q

2. (b) Write \(3 \ln 2 + \ln 15 − \ln 6\) as a single logarithm in the form \(\ln k\). [2]

   • M1 for the power law on the coefficient, then the product and quotient laws
   • A1 ln 20

Do not use a calculator for this question. [5]

1. (a) Find the exact value of \(\left(\frac{3}{5}\right)^{-3}\). [2]

   • M1 for flipping the fraction: \(\left(\frac{5}{3}\right)^{3}\)
   • A1 \(\frac{125}{27}\)

2. (b) Write \(\dfrac{(3x^4)^2}{18x^{-3}}\) in the form \(ax^p\). Write down the values of a and p. [3]

   • M1 for \((3x^4)^2=9x^{8}\)
   • M1 for dividing the numbers and subtracting the powers
   • A1 \(a = \frac{1}{2}\), p = 11

Solve each equation. [5]

1. (a) Solve \(\ln(x-2)=1.5\). Give x to 3 significant figures. [2]

   • M1 for \(x-2=e^{1.5}\)
   • A1 \(x = e^{1.5}+2 = 6.48\)

2. (b) Solve \(4^{x+1}=32^{x-2}\). Give the exact value of x. [3]

   • M1 for writing both sides as powers of 2: \(2^{2(x+1)}=2^{5(x-2)}\)
   • M1 for \(2x+2=5x-10\)
   • A1 x = 4

A water tank holds \(7.2\times10^{9}\) litres. A swimming pool holds \(3.0\times10^{5}\) litres. [5]

1. (a) Find how many pools the water in the tank would fill. Give your answer in the form \(a\times10^{k}\), where \(1\le a<10\) and \(k\in\mathbb{Z}\). [2]

   • M1 for \(\dfrac{7.2\times10^{9}}{3.0\times10^{5}}\) (\(2.4\times10^{4}\), the a values divided and the indices subtracted)
   • A1 \(2.4\times10^{4}\)

2. (b) A second tank holds \(9.6\times10^{8}\) litres. Find the total number of litres the two tanks hold. Give your answer in the form \(a\times10^{k}\). [2]

   • M1 for writing both numbers with the same power of 10, for example \(7.2\times10^{9}+0.96\times10^{9}\)
   • A1 \(8.16\times10^{9}\)

3. (c) One litre is \(10^{3}\) millilitres. Write down the volume of the first tank in millilitres, in the form \(a\times10^{k}\). [1]

   • A1 \(7.2\times10^{12}\) (\(7.2\times10^{9}\times10^{3}\))

The intensity I, in watts per square metre, of a sound at a level of L decibels satisfies \(\log_{10}I=-12+0.1L\). [5]

1. (a) Find the intensity of a sound at a level of 85 decibels. Give your answer in the form \(a\times10^{k}\), where \(1\le a<10\) and \(k\in\mathbb{Z}\), with a correct to 3 significant figures. [2]

   • M1 for \(\log_{10}I=-12+0.1\times85=-3.5\), so \(I=10^{-3.5}\)
   • A1 \(3.16\times10^{-4}\)

2. (b) A sound at 110 decibels and a sound at 80 decibels are compared. Find how many times more intense the 110 decibel sound is. Give your answer to 3 significant figures. [3]

   • M1 for \(\log_{10}I_{110}-\log_{10}I_{80}=0.1\times(110-80)=3\) (the \(-12\) cancels)
   • M1 for \(\log_{10}\dfrac{I_{110}}{I_{80}}=3\), so \(\dfrac{I_{110}}{I_{80}}=10^{3}\)
   • A1 1000

Solve each equation. [13]

1. (a) Solve \(\log_{10}(x+2) + \log_{10}4 = 2\). Give an exact answer. [3]

   • M1 for combining: log10 4(x + 2) = 2
   • M1 for 4(x + 2) = 10^2 = 100
   • A1 x = 23

2. (b) Solve \(\ln(3x+2) - \ln(x-3) = 2\). Give x to 3 significant figures. [4]

   • M1 for combining: ln ((3x + 2)/(x − 3)) = 2
   • M1 for (3x + 2)/(x − 3) = e^2
   • M1 for collecting the terms in x and solving
   • A1 5.51

3. (c) Solve \(3^{x+2} = 8^{x-1}\). Give x to 3 significant figures. [4]

   • M1 for taking ln of both sides: (x + 2) ln 3 = (x − 1) ln 8
   • M1 for expanding and collecting the terms in x
   • M1 for x = 2 ln 3 + 1 ln 8 over ln 8 − ln 3
   • A1 4.36

4. (d) Solve \(\log_{x}125 = \frac{3}{2}\). [2]

   • M1 for x^(3/2) = 125
   • A1 x = 25

The area N (m²) of a pond covered by algae, t days after it is first measured, is modelled by \(N = 30\times1.15^{t}\). [8]

1. (a) Write down the value of N when t = 0, and find the value of N when t = 5. [2]

   • A1 N = 30
   • A1 60.3 (30 × 1.15^5)

2. (b) Use your calculator to find when N reaches 150. Give t to 3 significant figures. [2]

   • M1 for solving 30 × 1.15^t = 150
   • A1 11.5

3. (c) A second model is \(M = 90\times1.05^{t}\). Find the time when M = N. Give t to 3 significant figures. [2]

   • M1 for solving 30 × 1.15^t = 90 × 1.05^t
   • A1 12.1

4. (d) Find the time for N to multiply by 10. Give t to 3 significant figures. [2]

   • M1 for solving 1.15^t = 10
   • A1 16.5

An ecologist studies how the number of species y found on an island depends on its area x (km²). The areas in the study range from 0.02 km² to 20 000 km². [13]

1. (a) Write down the logarithm to base 10 of each of the values 0.04, 15, 8000. Give each to 3 significant figures. [2]

   • M1 for using log10 of the values
   • A1 −1.40, 1.18, 3.90

2. (b) Give one reason why a logarithmic scale is useful for data like this. [1]

   • R1 the values cover several powers of 10, so the logs are of similar size and can be shown on one graph

3. (c) For the data in the table, the correlation coefficient for \(\log_{10}y\) against x is r = 0.7615, and for \(\log_{10}y\) against \(\log_{10}x\) it is r = 0.9983. Decide whether an exponential model or a power model fits the data better. Give a reason. [2]

   • M1 for comparing the two values of r (the one closer to 1 is nearer a straight line)
   • A1 a power model, \(y=a x^k\), because log y against log x is the straighter graph

4. (d) The line of best fit for \(\log_{10}y\) against \(\log_{10}x\) has gradient 0.298 and intercept 0.708. Write the model in the form \(y=a x^k\). [3]

   • M1 for k = gradient = 0.298
   • M1 for a = 10^intercept = 10^0.708
   • A1 y = 5.11x^0.298

5. (e) Use the model to estimate y when x = 250. [2]

   • M1 for substituting x = 250
   • A1 26.5

6. (f) A different relationship has the form \(y=a\times b^{x}\). The graph of \(\log_{10}y\) against x is a straight line through \((0, 0.7)\) and \((3, 1.9)\). Find a and b. [3]

   • M1 for gradient = (1.9 − 0.7) ÷ (3 − 0) = 0.400
   • M1 for b = 10^gradient and a = 10^intercept (intercept 0.700)
   • A1 a = 5.01, b = 2.51

The concentration C mg/L of a drug in a patient's blood, t hours after a dose, is modelled by \(C=60\times0.78^{t}\). [7]

1. (a) Write down the value of C when \(t=0\), and find the value of C when \(t=5\). Give your answer to 3 significant figures. [2]

   • A1 60
   • A1 17.3 (\(60\times0.78^{5}\))

2. (b) Write down the percentage of the C lost each hour. [1]

   • A1 22% (0.78 means a decrease of 22%)

3. (c) Use your calculator to find when \(C=25\). Give your answer to 3 significant figures. [2]

   • M1 for solving \(60\times0.78^{t}=25\)
   • A1 t = 3.52

4. (d) The drug is safe once the concentration is less than 2 mg/L. Find the first whole number of hours after which it is safe. [2]

   • M1 for solving \(60\times0.78^{t}=2\) (t = 13.7)
   • A1 14 (the first whole number above 13.7)

A yeast culture grows so that its mass M, in milligrams, after t hours is modelled by \(M=M_0e^{kt}\), where \(M_0\) and k are constants. At \(t=0\) the mass is 800 mg. At \(t=5\) the mass is 2600 mg. [8]

1. (a) Find the value of k. Give your answer to 3 significant figures. [3]

   • M1 for \(800e^{5k}=2600\), so \(e^{5k}=\dfrac{13}{4}\)
   • M1 for \(5k=\ln\dfrac{13}{4}\)
   • A1 k = 0.236

2. (b) Find the time at which the mass is 15 000 mg. Give your answer to 3 significant figures. [2]

   • M1 for \(800e^{kt}=15\,000\), so \(e^{kt}=\dfrac{75}{4}\), and \(t=\dfrac{\ln(75/4)}{k}\)
   • A1 t = 12.4

3. (c) At \(t=14\) a fungicide is added. From then the mass is modelled by \(M=15\,000\,e^{-0.42s}\), where s is the time in hours since the fungicide was added. Find the half-life of this decay. Give your answer to 3 significant figures. [3]

   • M1 for \(15\,000e^{-0.42s}=7500\), so \(e^{-0.42s}=0.5\)
   • M1 for \(-0.42s=\ln0.5=-\ln2\), so \(s=\dfrac{\ln2}{0.42}\)
   • A1 1.65 hours

The number of species S found on an island of area x square kilometres is modelled by \(S=a+b\ln x\), where a and b are constants. An island of area 3 km² has 4.1 species per sampling plot on average and an island of area 20 km² has 8.7. [8]

1. (a) Find the value of b and the value of a. Give your answers to 3 significant figures. [4]

   • M1 for \(a+b\ln3=4.1\) and \(a+b\ln20=8.7\)
   • M1 for subtracting: \(b(\ln20-\ln3)=4.6\), so \(b=\dfrac{4.6}{\ln(20/3)}\)
   • A1 b = 2.42
   • A1 a = 1.44 (using the full value of b)

2. (b) Use the model to predict the number of species per plot on an island of area 45 km². Give your answer to 3 significant figures. [2]

   • M1 for \(S=a+b\ln45\)
   • A1 10.7

3. (c) Use the model to find the area of an island with 12 species per plot. Give your answer to 3 significant figures. [2]

   • M1 for solving \(a+b\ln x=12\)
   • A1 78 km²